Properties of Exponents in Exponential Functions

Gain a deep understanding of zero and negative exponents, along with the key properties of exponents. Learn how to simplify expressions involving exponents and apply the product, quotient, and power of powers properties. Examples and practice problems included.

• Exponents
• Exponential Functions
• Zero Exponents
• Negative Exponents
• Properties

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Uploaded on Jul 29, 2024 | 4 Views

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1. Chapter 6 Exponential Functions 6.1 Properties of Exponents How can you write general rules involving properties of exponents? 1. Students will understand and be able to use zero and negative exponents. 2. Students will know and be able to use the properties of exponents.

2. 1. Students will understand and be able to use zero and negative exponents. Zero Exponent For any nonzero number a, a0 = 1. The power 00 is undefined. a a = = 0 0 0 , where 1 4 1 Negative Exponents For any integer n and any nonzero number a, a-nis the reciprocal of an. 1 4 4 a 1 = = n a 2 a 0 , where n 2

3. 1. Students will understand and be able to use zero and negative exponents. Examples Evaluate each expression. 1 = 1 ( 2) 4 0 6.7 = ( 2) 4 1 = 16 1 = = = = = 0 0 2 5 1 5 11 1 3 9 3 2

4. You try! = 1 ( 3) 3 1 0 12 = ( 3) 3 1 27 = = 1 ( ) 1 = 2 0 5 9 5 2 = 25

5. You try! = = 1 1 1 24 2 0 4 24 = 4 2 1 0 = Because 16 4 4 3 3 2 2 = 3 4 4 3 = 1 = 0 356 16 9 =

6. Simplify the expression. 4 1 y 0 x x 4x y 2 5 2 5 3 3 = = 3 3 0 1 9 9 x = 3 4 1 y = 5 = 3 4y

7. You Try! 2 9 8 7 q p p 11 0 11 1 10 r s r s = = 8 2 9 7 q 2 9 s r 3 9 49q p = = 11 8 9

8. 2. Students will know and be able to use the properties of exponents. Product of Powers Property To multiply powers with the same base, add their exponents. m n + 6 3 + = 4 4 = = m n 6 3 9 4 a a a 4 Quotient of Powers Property To divide powers with the same base, subtract their exponents. 6 4 4 m a a 6 3 = = 3 4 4 m n = a 0 a , where 3 n Power of a Powers Property To find a power of a power, multiply the exponents. = = m n m n mn = = ( ) 6 3 a a a 18 6 3 (4 ) 4 4

9. 2. Students will know and be able to use the properties of exponents. Examples Simplify each expression. Write your answer using only positive exponents. 4 3 2 6 + 3 3 = 4 ( 3) 2 6 ( ) = z 2 3 ( 4) ( 4) z z 1 z 2 7 ( 4) = 7 = = 83 12 ( 4) = 5 = 1 12 = ( 4) 5 1 = 1024

10. You Try! ( 2) ( 1) 6 = x 9 ( 9) + 4 ( 6) + 10 10 2 1 = = = 9 9 4 6 (6 ) 10 10 x x = = 0x 2 = 2 6 1 1 = 36 = 2 10 1 100 =

11. You Try! 9 5 5 3 ( 2) (2 ) = = 9 7 8 ( 7) 3+ 3 3 3 2 = = 8 7 2 2 5 7 = 13 6 = = 2 5 = 3 1 2 25 = 6 1 64 =

12. 2. Students will know and be able to use the properties of exponents. Power of a Product Property To find a power of a product, find the power of each factor and multiply. ( ) ( ) 4 m 3 2 = = 81 16 m m = 4 4 ab a b 3 2 Power of a Quotient Property To find the power of a quotient, find the power of the numerator and the power of the denominator and divide. 27 8 3 3 3 2 3 2 m m a b a b = = = b 0 , where 3 m

13. 2. Students will know and be able to use the properties of exponents. Simplify each expression. Write your answer using only positive exponents ( ) ( ) 5 = = = = 3 3 3 3 5 3 5 ) 15 5 x 125x 2 ( 32x x 5x 3 2x 4 4 4 4 3 81 16 d d 243 32x 3 d 5 5 5 3 2 3 x = = = = = 4 2 2 5 5 5 3 2x 2 x

14. You Try!!! Simplify each expression. Write your answer using only positive exponents 1 (10 ) y ( ) 1 1 3 10y = = = 3 3 3 3 10 y 1000y ( ) = 1 3 x = 3 3 3 3 3 27x 3x 1024 n 5 5 4 n 4 n = = 5 1 5 5 49 2 2 2 7 5 7 6 x = = = 2 5 2 10 6 ( ) x 5 7 36x 6x

15. Lets Review 1. Students will understand and be able to use zero and negative exponents. Any number, but zero, raise to the zero power is 1 Negative exponents mean reciprocals. 2. Students will know and be able to use the properties of exponents. When you multiply the same base you add the exponents. When you divide the same base you subtract the exponents. A power raised to a power, you multiply the exponents. You can distribute exponents across multiplication and division only.